Refinements of the Littlewood-Richardson Rule

James Haglund, Kurt W. Luoto, Sarah Mason, Stephanie van Willigenburg

Published 2009-08-25, updated 2009-12-04Version 3

In the prequel to this paper, we showed how results of Mason involving a new combinatorial formula for polynomials that are now known as Demazure atoms (characters of quotients of Demazure modules, called standard bases by Lascoux and Schutzenberger) could be used to define a new basis for the ring of quasisymmetric functions we call Quasisymmetric Schur functions (QS functions for short). In this paper we develop the combinatorics of these polynomials futher, by showing that the product of a Schur function and a Demazure atom has a positive expansion in terms of Demazure atoms. As a by-product, using the fact that both a QS function and a Demazure character have explicit expressions as a positive sum of atoms, we obtain the expansion of a product of a Schur function with a QS function (Demazure character) as a positive sum of QS functions (Demazure characters). Our formula for the coefficients in the expansion of a product of a Demazure character and a Schur function into Demazure characters is similar to known results and includes in particular the famous Littlewood-Richardson rule for the expansion of a product of Schur functions in terms of the Schur basis.